MATH 436 Notes: Finitely generated Abelian groups

Definition 1.1 (Direct Products). Let {Gα}α∈I be a collection of groups indexed by an index set I. We may form the Cartesian product ∏ α∈I Gα. The elements of this Cartesian product can be denoted by tuples (aα)α∈I . We refer to the entry aα as the αth component of this tuple. We define a multiplication on this Cartesian product componentwise, i.e., (aα) ⋆ (bα) = (aα ⋆α bα) where ⋆α is the group multiplication in Gα. It is easy to check that this makes ∏ α∈I Gα into a group with identity element (eα)α∈I where eα is the identity element of Gα for each α ∈ I. When the index set I = {1, 2, . . . , n} is finite we usually write ∏ α∈I Gα as ∏n i=1 Gi or sometimes more simply as G1 × · · · × Gn. Finally when the index set I = N we write ∏ α∈N Gα as ∏∞ i=0 Gi.